Adjoint based optimal control using mesh-less discretizations
- An easy numerical handling of time-dependent problems with complicated geometries, free moving boundaries and interfaces, or oscillating solutions is of great importance for many applications, e.g., in fluid dynamics (free surface and multiphase flows, fluid-structure interactions [22, 18, 24]), failure mechanics (crack growth and propagation ), magnetohydrodynamics (accretion disks, jets and cloud simulation ), biophysics and -chemistry. Appropriate discretizations, so-called mesh-less methods, have been developed during the last decades to meet these challenging demands and to relieve the burden of remeshing and successive mesh generation being faced by the conventional mesh-based methods, [16, 10, 3]. The prearranged mesh is an artificial constraint to ensure compatibility of the mesh-based interpolant schemes, that often conflicts with the real physical conditions of the continuum model. Then, remeshing becomes inevitable, which is not only extremely time- and storage consuming but also the source for numerical errors and hence the gradual loss of computational accuracy. Apart from this advantage, mesh-less methods also lead to fundamentally better approximations regarding aspects, such as smoothness, nonlocal interpolation character, flexible connectivity, refinement and enrichment procedures, . The common idea of mesh-less methods is the discretization of the domain of interest by a finite set of independent, randomly distributed particles moving with a characteristic velocity of the problem. Location and distribution of the particles then account for the time-dependent description of the geometry, data and solution. Thereby, the global solution is linearly superposed from the local information carried by the particles. In classical particle methods [20, 21], the respective weight functions are Dirac distributions which yield solutions in a distributional sense.